Birthday Probabilities

The paradox of birthdays is a mathematical problem put forward by Von Mises, who looks for the value N in the problem: In a group of N people there is 50% chance that at least 2 people in the group share the same birthday (day + month). The answer is N = 23, which is quite counterintuitive, hence the paradox.

During the calculation of the birthdate paradox, it is supposed that births are equally distributed over the days of a year (it is not exactly true in reality). In the following, a year has 365 days (leap years are ignored).

Probability is 1/365 = 0.0027 = 0.27%, indeed, one chance out of 365 to be bord a precise day.

The probability is 364/365 = 0.9973 = 99.73%, indeed, he must be born a distinct day of mine, so there is 364 possibilities out of 365.

This calculation is the same as What is the probability for a person to be born a given day of the year? This given day is my birthday.

Example: 1/365 = 0.0027 = 0.27%

The calculation could also be formulated by noting that the probability for a person to be born the same day as me is the opposite as the probability for a person to be born a different day than mine.

Example: 1 - 364/365 = 1 - 0.9973 = 0,0027 = 0.27%

It is also the probability that a mother has a second child born on the same day (not the same year) as the first.

This calculation is the same as What is the probability for a person (the child) to be born a given day of the year (the mother birthday)?.

Example: 1 chance in 365 or P = 0.0027 or 0.27%

The probability that a mother and her 2 children were born on the same day

Example: 1 chance in 365 * 365 = 133225 or P = 0.0000075 or 0.00075%

The probability that a mother and her N children were born on the same day is 1 in 365 ^ N

$$ P(X) = 1/365^N $$

For N = 2, it is the same as probability for a person to be born the same day as me or the opposite as the probability for a person to be born a different day than mine.

Example: P(N=2) = 1 - (364/365) = 0,0027 = 0.27%

For other people, they have to be born also a distinct day than me, but also a distinct day of the other one (then calculate the opposite):

Example: $$ P(N=3) = 1 - (364/365) * (363/365) = 0,0082 = 0.82\% \\ P(N=4) = 1 - (364/365) * (363/365) * (362/365) = 0,0164 = 1.64\% \\ P(N=5) = 1 - (364/365) * (363/365) * (362/365) * (361/365) = 0,0271 = 2.71\% $$

About 0.5, 50% (1 chance out of 2)

366 are needed (367 if leap years are taken into account). Indeed, every day of the year plus one are needed to be sure that at least a couple of two people share the same birthday.

For a given date, 253 people are needed to get a probability of 50% that 2 people to be born a same precise day.

Probabilities are multiplied

Example: P(N=2) = 1 - (364/365) = 0,0027 = 0.27%

Example: P(N=3) = 1 - (364/365)^2 = 0,0055 = 0.55%

Example: P(N=n) = 1 - (364/365)^(n-1)

The probability is

Example: P = (364/365)^N

With the hypothesis of a world population of 8 billion people, there are 8000000000 / 365.25 or about 22 million people born on a given day / month.

Limiting to USA (350 millions inhabitants), there are 350000000 / 365.25 or nearly 1 millions US people born on a given day / month date.